An improvement on the bound for the acyclic chromatic index

2026-02-16

AI Summary

This study addresses the problem of acyclic edge coloring of graphs, which seeks to minimize the number of colors assigned to edges such that no cycle is bichromatically alternating. The work introduces, for the first time, unordered (non-planar) trees as witness structures within the framework of the Lovász Local Lemma and combines this insight with analytic combinatorial techniques to obtain a more precise estimation of the dependency structure among bad events. This novel approach significantly improves the upper bound on the acyclic chromatic index for graphs of maximum degree Δ, reducing it from the previous best-known bound of 4(Δ−1) to 3.142(Δ−1)+1, thereby establishing a new state-of-the-art result in the field.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $\Delta$, the acyclic chromatic index is at most $3.142(\Delta-1)+1$, improving on the (best to date) bound of Fialho et al. (2020). Our improvement is made possible by considering unordered (non-plane) trees, instead of ordered (plane) ones, as witness structures for the Lov\'{a}sz Local Lemma, a key combinatorial tool often used in related works. The counting of these witness structures entails methods of Analytic Combinatorics.

Problem

• acyclic chromatic index
• edge coloring
• graph theory
• Lovász Local Lemma
• combinatorics

Innovation

• acyclic chromatic index
• Lovász Local Lemma
• unordered trees
• Analytic Combinatorics
• graph coloring